082 



{G*i],) we fmd the intersections B„ A^ and A^ o( g„ h,^* and h,,*. 



To the points F of the line / lying in ff correspond the pairs of 

 points P, P" lying on the curve of the fifth order, which ff has 

 moreover in common with the surface A"; this curve passes through 

 the points Ak, Ak^' and has the points Bk as nodes." 



So ive find a cubic involution possessing the same properties as 

 the cubic involution (P') considered before. 



11. We are now going to consider the case that the plane (p is 

 laid through a straight line c, resting on g„<u,g^ and cutting these 

 lines in the points B„ B„ B^. The three hyperboloids //determined 

 by these points have the line c in common besides a conic (p^ through 

 G,(P, resting on c,g,,g^ and g^ and forming with c a curve of 

 the [y']. For the conies passinu- through (V, Cr* and cutting (/i, //,, r/,, 

 form a surface of the fourth order, cut by c in a point not lying 

 on one of the lines g. The three hyperboloids mentioned cut (f along 

 three lines b„b„b^, meeting in a point C not lying on c, whereof' 

 intersects the plane (p again. 



The curves [r/'] passing through B^, meet (p in the pairs of points 

 P',F', of an involuticm on b^. So Bk are now^ singular points oH the 

 first order. C too is a singidar point now ; for the figure (r/*, c) has 

 all the points of c in common with </, so that each pair of c corre- 

 sponds to C. 



The conic {B^f of tiie general case has been replaced here by the 

 pair of lines {b^,c); on b^ lie now the singular points A^,A*. 



The singular points and lines now form a configuration (10,, lOg), 

 viz. the well-known configuration of Desargues. For in the lines 

 b,,b„b„ passing through C\ the triangles .4, .4, .4, and A*A^*A^* 

 are inscribed, the pairs of corresponding sides a,*, a, ; a,*, a, ; a,*, a, 

 of which meet in the collinear points B^, B,, B^. 



From the curve (/\ which in the general case corresponds to a 

 line r, the line c falls away ; in connection with this the curve of 

 coincidences y^ passes into a conic. 



On the o' with one node D, now associated to r, exists only o?2e 

 involution of pairs ; the points P, P", which form triangles of 

 involution with the points of r, lie therefore on the lines p passing 

 through £>; consequently n=:l. 



This involution differs from the (P') described by Reye only in 

 this respect that the singular point C does not correspond to the 

 pairs of an /•' on c, as all the points of c have been associated to C. 



12. Another (P') differing in this respect from the involution 



